Cantor’s paradox

Cantor’s paradox demonstrates that there can be no largest cardinality. In particular, there must be an unlimited number of infinite cardinalities. For suppose that α were the largest cardinal. Then we would have |𝒫⁢(α)|=|α|. (Here 𝒫⁢(α) denotes the power set of α.) Suppose f:α→𝒫⁢(α) is a bijection proving their equicardinality. Then X={β∈α∣β∉f⁢(β)} is a subset of α, and so there is some γ∈α such that f⁢(γ)=X. But γ∈X↔γ∉X, which is a paradox.

The key part of the argument strongly resembles Russell’s paradox, which is in some sense a generalization of this paradox.

Besides allowing an unbounded number of cardinalities as ZF set theory does, this paradox could be avoided by a few other tricks, for instance by not allowing the construction of a power set or by adopting paraconsistent logic.